NEW YORK — Hidden from the public in his Manhattan high-rise, Donald Trump huddled Tuesday with Vice President-elect Mike Pence as he tried to fill out key posts in his Cabinet. But the transition team appeared to be straining under the enormous challenge of setting up an administration.

Former Rep. Mike Rogers, a respected Republican voice on national security issues, announced he was stepping down from the transition effort. And an apparent clerical oversight effectively halted the Trump team’s ability to coordinate with President Barack Obama’s White House.

With Trump out of sight now for several days, his allies engaged in an unusual round of public speculation about his potential appointments — including their own futures — as the president-elect and his aides weighed the nation’s top national security posts.

Former New York City Mayor Rudy Giuliani seemed to be angling for secretary of state, and a Trump official said the job was his if he wanted it.

However, a second official cautioned that John Bolton, a former U.S. ambassador to the United Nations, remained in contention. The officials weren’t authorized to speak publicly on the matter and demanded anonymity.

In Washington late Monday, Giuliani said Bolton would be a “very good choice.” Asked if there was anyone better, Giuliani replied: “Maybe me, I don’t know.”

Businessman Carl Icahn took to Twitter to disclose details of a conversation he said he had with the president-elect. Icahn said Trump was considering Steve Mnuchin, a former Goldman Sachs banker, and Wilbur Ross, a billionaire investor, to lead the Treasury and Commerce departments.

Pence, now running the transition team, ignored questions from reporters as he walked through the lobby of Trump Tower, a thick binder tucked under his arm. He took over from New Jersey Gov. Chris Christie, who spent months running transition operations before his demotion last week.

The switch has slowed Trump’s ability to coordinate with the White House. Pence has yet to sign a memorandum of understanding facilitating interactions between his team and Obama administration officials. Christie had signed the document, but Pence’s promotion makes it no longer valid.

A stream of Trump advisers arrived Tuesday at his eponymous skyscraper to plan for the next administration.

“He’s going to be a great president. He’s got a team,” said Ret. Lt. Gen. Keith Kellogg, who is advising on defense issues. “He’s got great people. They believe in him. They believe in the republic. It’s going to be just fine.”

The team appeared to be focusing on filling out the top national security jobs. Trump’s selections will be the first signals to anxious international allies about the direction he plans to take U.S. foreign policy.

Giuliani, 72, would be an out-of-box choice for secretary of state. A former mayor and federal prosecutor, he is known for his hard line law-and-order views.

Bolton has years of U.S. foreign policy experience, but has raised eyebrows with some of his hawkish stances, including a 2015 New York Times op-ed in which he advocated bombing Iran to halt the country’s nuclear program.

Alabama Sen. Jeff Sessions, a loyal Trump ally and immigration hardliner, is said to be a contender to lead the Pentagon as defense secretary.

Trump also is considering Richard Grenell as U.S. ambassador to the United Nations, which would bring some experience and diversity to his nascent administration. Grenell, who served as U.S. spokesman at the U.N. under President George W. Bush, would be the first openly gay person to fill a Cabinet-level foreign policy post.

The transition planning comes amid continued backlash from Trump’s weekend decision to appoint Steve Bannon, a man celebrated by the white nationalist movement, to serve as his chief strategist and senior adviser.

“After winning the presidency but losing the popular vote, President-elect Trump must try to bring Americans together — not continue to fan the flames of division and bigotry,” House Democratic leader Nancy Pelosi said. She called Bannon’s appointment “an alarming signal” that Trump “remains committed to the hateful and divisive vision that defined his campaign.”

Until joining Trump’s campaign this summer, Bannon led the Breitbart website that appealed to the so-called “alt-right” — a movement often associated with far right efforts to preserve “white identity,” oppose multiculturalism and defend “Western values.”

Breitbart frequently has targeted House Speaker Paul Ryan. Asked about Bannon’s appointment, Ryan said, “I would just simply say the president will be judged on results.”

On Monday, Trump spoke for the first time by telephone to Russian President Vladimir Putin.

Trump “is very much looking forward to having a strong and enduring relationship with Russia and the people of Russia,” his transition office said. Trump, who has no foreign policy experience, also has spoken in recent days with the leaders of China, Mexico, South Korea and Canada.

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